सिद्ध कीजिए कि दो समकोण अतिपरवलय के प्रतिच्छेद बिन्दुओं से गुजरने वाला शंकु भी समकोण अतिपरवलय होता है।
See Answer →शांकव x2 - 2xy + y² 3x + 2y + 3 = 0 को अनुरेखित कीजिए।
See Answer →जांच कीजिए कि निम्नलिखित कथन सत्य हैं अथवा असत्य। अपने उत्तर की पुष्टि लघु व्याख्या या प्रति-उदाहरण द्वारा कीजिए।
(i) संख्याएँ एक रेखा की दिक्कोज्याएं हैं।
ii) बिंदु (1,2), (7,6) और (4,4) संरेखीय हैं।
(iii) शांकव 12x2 + 12xy + 3y2 + 2x + y = 0 पतित है।
iv) दीर्घवृत्त का समतल वाइ = 5 से प्रतिच्छेद एक वृत्त है।
(v) शांकवज 3x2 + y² + 2xy+x-y-z+ 1 = 0 अकेंद्रीय है।
(vi) रेखा y = x परवलय y² = cx,c, > 0 की स्पर्श रेखा है।
(vii) समीकरण 2x² + y + z + 1 = 0 एक परवलज को निरूपित करता है।
(viii) किसी रेखा-खण्ड का उसकी लंब रेखा पर प्रक्षेप उस रेखा-खण्ड की लंबाई के बराबर होता है।
(ix) रेखाएं x = y, z = 2 और x = y, z = 0 परस्पर प्रतिच्छेद करती हैं।
(x) बेलन का समतल परिच्छेद एक वृत्त होता है।
See Answer →Find the equation of tangent plane to the conicoid x² + 3y² = 4z at (2, -4, 13). Represent the tangent plane geometrically.
See Answer →Identify and trace the conicoid y² + 3z2 = x. Describe its sections by the planes y = 0 and z = 0.
See Answer →Find the projection of the line segment joining the points (1,-1, 6) and (4, 3, 2) on the line x-4/ 3 = -y = z/5.
See Answer →Find the transformation of the equation 12x² - 2y2 + z² = 2xy if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are 1, -3, 0; 3, 1, 0; 0, 0, 1.
See Answer →Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.
(i) x² + y² + z² + 4x + 3y – z = 0
(ii) 2x2-y2z2 + xy + yz - zx = 1
(iii) x2 + y2 - z2 -2xy -3yz - 6zx + x - 2y + 5z + 4 = 0
See Answer →Show that the conicoid 2x² + 2y² + xyyz + zx + 2xy + 5z + 1 = 0 is central. Hence find its centre.
See Answer →Transform the equation x² + 2y² 6z2 - 2x - 8y+3 = 0 by shifting the origin to (1, 2, 0) without changing the directions of the coordinate axes. What object does this new equation represent? Give a rough sketch of it.
See Answer →Show that the perpendiculars drawn from the origin to tangent planes to the cone x2 y2 + 5z² + 4xy = 0 lie on the cone x2 y2 + z² + 4xy = 0.
See Answer →Find the equation of the cylinder with base x² + y² + z²-3x6z + 9 = 0, x - 2y+2z-6 = 0.
See Answer →Find the angle between the lines of intersection of the cone 4x2 + y² + 4z² + 4yz + 2zx = 0 and the plane x + 2y + 3z = 0.
See Answer →Find the equation of the sphere touching the plane 8x + 5y + 3z + 1 = 0 at (3,-1,-1) and cutting the sphere x² + y2 + z²-2x+y-z-6=0 orthogonally.
See Answer →Show that the plane 2x + y + 2z = 0 is a tangent plane to the sphere x² + y² + z2-2x+2y-2z + 2 = 0.
See Answer →Find the distance of the origin from the plane which passes through (2, 1, 8), (1, 0, 2) and (-3, 4, 6).
See Answer →Find the equation of the plane which passes through the line of intersection of the planes 3x + 4y - 5z = 9 and 2x+6y+6z = 7 and which is perpendicular to the plane 3x + 2y5z + 6 = 0.
See Answer →Find the equations of the line through (1,3,4) and parallel to the line joining the points (-4, 5, 3) and (8, 9, 7).
See Answer →Prove that the length of the chord of a parabola which passes through the focus and which is inclined at 30° to the axis of the parabola is four times the length of the latus rectum.
See Answer →Find the eccentricity, foci, centre and directrices of the ellipse . Also 4 give a rough sketch of it.